find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=\fra…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{1}{(x + h)-1}=\frac{1}{x+h - 1}$

Step2: Calculate $f(x + h)-f(x)$

$f(x + h)-f(x)=\frac{1}{x + h-1}-\frac{1}{x - 1}=\frac{(x - 1)-(x + h - 1)}{(x - 1)(x + h-1)}=\frac{x-1-x - h + 1}{(x - 1)(x + h-1)}=\frac{-h}{(x - 1)(x + h-1)}$

Step3: Calculate the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{\frac{-h}{(x - 1)(x + h-1)}}{h}=\frac{-h}{(x - 1)(x + h-1)}\times\frac{1}{h}=-\frac{1}{(x - 1)(x + h-1)}$

Answer:

$-\frac{1}{(x - 1)(x + h-1)}$